The average performed in (REF ) is actually analogue to a first-order Lie serie expansion and the averaging process can be seen as a periodic change of variable. Indeed, noting \(x\)  and \(x^{\prime }\)  the variables of the Hamiltonian respectively before and after the average, one has \(x=e^{L_W}x^{\prime }\)  where \(L_W=\left\lbrace W,\cdot \right\rbrace \)  denotes the total time derivative along the trajectories of the scalar field \(W\) , which is constrained by the cohomological equation [1]}
\(\left\lbrace H_K,W\right\rbrace =H_P-\frac{1}{2\pi }\int _0^{2\pi }H_P\,d\xi _2.\) 
